Vibration of Rectangular Plates by the Ritz Method

1950 ◽  
Vol 17 (4) ◽  
pp. 448-453 ◽  
Author(s):  
Dana Young
Keyword(s):  
Ritz Method ◽  
Normal Modes ◽  
Approximate Solutions ◽  
The Other ◽  
Rectangular Plates ◽  
Elastic Plates ◽  
Modes Of Vibration ◽  
Ritz’S Method ◽  
Square Plate ◽  
Set Up

Abstract Ritz’s method is one of several possible procedures for obtaining approximate solutions for the frequencies and modes of vibration of thin elastic plates. The accuracy of the results and the practicability of the computations depend to a great extent upon the set of functions that is chosen to represent the plate deflection. In this investigation, use is made of the functions which define the normal modes of vibration of a uniform beam. Tables of values of these functions have been computed as well as values of different integrals of the functions and their derivatives. With the aid of these data, the necessary equations can be set up and solved with reasonable effort. Solutions are obtained for three specific plate problems, namely, (a) square plate clamped at all four edges, (b) square plate clamped along two adjacent edges and free along the other two edges, and (c) square plate clamped along one edge and free along the other three edges.

Bending of Clamped Plates

1947 ◽  
Vol 14 (1) ◽  
pp. A55-A62
Author(s):  
W. B. Stiles
Keyword(s):  
Exact Solution ◽  
Normal Modes ◽  
Point Load ◽  
Simultaneous Equations ◽  
Central Point ◽  
The Other ◽  
Rectangular Plates ◽  
Vibrating Membrane ◽  
Square Plate ◽  
Infinite Set

Abstract The exact solution of thin rectangular plates clamped on all or part of the boundary requires the solution of two infinite sets of simultaneous equations in two sets of unknowns. A method of obtaining an approximate solution based upon minimization of energy and requiring the solution of the first i equations of a single infinite set of simultaneous equations is described and illustrated in this paper. The approximation functions are derived from functions representing the normal modes of a freely vibrating membrane for the same region. Solutions are obtained for a rectangular clamped plate supporting a uniform or a central point load and for a square plate clamped on two adjacent edges and pinned on the other two edges with either a uniform or a central point load. Analytical results are compared with experimentally determined deflections and stresses.

Vibration of Rectangular and Skew Cantilever Plates

1951 ◽  
Vol 18 (2) ◽  
pp. 129-134 ◽  
Author(s):  
M. V. Barton
Keyword(s):  
Ritz Method ◽  
Normal Modes ◽  
Approximate Solutions ◽  
Equal Length ◽  
Uniform Beam ◽  
Analytical Results ◽  
Modes Of Vibration ◽  
Experimental Values ◽  
Skew Angles ◽  
Plate Deflection

Abstract The Ritz method is used to determine approximate solutions for the frequencies and modes of vibration of uniform rectangular and skew cantilever plates. The functions used to represent the plate deflection correspond to those which define the normal modes of vibration of a uniform beam. Solutions are obtained for (a) uniform rectangular cantilever plates with cantilever span-to-breadth ratios of 1/2, 1, 2, and 5; (b) uniform skew cantilever plates with sides of equal length and skew angles of 15 deg, 30 deg, and 45 deg. Experimental values are presented and the correlation with analytical results discussed.

Flexural Vibration of Rectangular Plates with Stiffeners Parallel to the Edges

1963 ◽  
Vol 67 (634) ◽  
pp. 664-668 ◽  
Author(s):  
S. Mahalingam
Keyword(s):  
Natural Frequencies ◽  
Ritz Method ◽  
Flexural Vibration ◽  
Approximate Solutions ◽  
The Other ◽  
Rectangular Plates ◽  
Equivalent System ◽  
Simply Supported ◽  
Finite Difference Equations ◽  
Four Corners

SummaryThe basis of the procedure described in the paper is the replacement of the stiffeners by an approximately equivalent system of line springs. One of two methods may then be used to determine the natural frequencies. A rectangular plate with edge stiffeners, point-supported at the four corners, is used to demonstrate the application of the Rayleigh-Ritz method. Numerical results obtained are compared with known approximate solutions based on finite difference equations. A Holzer-type iteration is employed in the case of a plate with parallel stiffeners, where the two edges perpendicular to the stiffeners are simply supported, the other two edges having any combination of conditions.

scholarly journals On the figures obtained by strewing sand on vibrating surfaces, commonly called acoustic figures

1837 ◽  
Vol 3 ◽  
pp. 180-181
Keyword(s):  
The Other ◽  
Vertical Line ◽  
Rectangular Plates ◽  
Regular Form ◽  
Modes Of Vibration ◽  
The Subject ◽  
Square Plate

The author, after adverting to the imperfect notice taken by Gali­leo and by Hooke of the phenomena which form the subject of this paper, ascribes to Chladni exclusively the merit of the discovery of the symmetrical figures exhibited by plates of regular form when made to sound. He proposes a notation, by means of two numbers separated by a vertical line, for expressing the figures resulting from the vibrations of square or rectangular plates. He gives a table of the relative sounds expressed both by their musical names and by the number of their vibrations, of all the modes of vibration of a square plate, as ascertained by the experiments of Chladni. He then pro­ceeds to class and analyse the various phenomena observed under these circumstances, and shows that all the figures of these vibrating surfaces are the resultants of very simple modes of oscillation, occurring isochronously, and superposed upon one another; the resultant figure varying with the component modes of the vibration, the num­ber of the superpositions, and the angles at which they are superposed. In the present paper, which forms the first part of his investigation, he confines himself to the figures of square and other rectangular plates. The author finds that the principal results of the superposition of two similar modes of vibration are the following :—first, the points where the quiescent lines of each figure intersect each other remain quiescent points in the resultant figure; secondly, the quiescent lines of one figure are obliterated, when superposed, by the vibratory parts of the other; thirdly, new quiescent parts, which may be called points of compensation, are formed whenever the vibrations in opposite directions neutralize each other; and, lastly, at other points, the mo­tion is as the sum of the concurring, or the differences of the opposing vibrations at these points. After considering various modes of binary superposition, the author examines the cases of four co-existing superpositions.

The Normal Modes of Vibrations of Beams Having Noncollinear Elastic and Mass Axes

1940 ◽  
Vol 7 (3) ◽  
pp. A97-A105
Author(s):  
Clyne F. Garland
Keyword(s):  
Physical Properties ◽  
Natural Frequencies ◽  
Ritz Method ◽  
Normal Modes ◽  
Longitudinal Axis ◽  
Cantilever Beams ◽  
Amplitude Ratios ◽  
Numerical Example ◽  
Modes Of Vibration ◽  
Axis Passing

Abstract This analysis deals with vibration characteristics of cantilever beams in which the longitudinal axis, passing through the mass centers of the elementary sections, is not collinear with the longitudinal axis about which the beam tends to twist under the influence of an applied torsional couple. Expressions are derived from which the natural frequencies and normal modes of vibration of such a beam can be determined. The Rayleigh-Ritz method is employed to determine the frequencies and amplitude ratios. Following the development of the general expressions, more specific equations are derived which express the natural frequencies and relative amplitudes of motion in each of two normal modes of vibration. The theoretical relationships of the several physical properties of the beam to the natural frequencies of vibration are shown graphically. Finally a numerical example is presented for a particular beam, and the computed natural frequencies and normal modes are compared with those determined experimentally.

scholarly journals Boundary Characteristic Orthogonal Polynomials in the Study of Transverse Vibrations of Nonhomogeneous Rectangular Plates with Bilinear Thickness Variation

2012 ◽  
Vol 19 (3) ◽  
pp. 349-364 ◽  
Author(s):  
R. Lal ◽  
Yajuvindra Kumar
Keyword(s):  
Orthogonal Polynomials ◽  
Ritz Method ◽  
Cartesian Product ◽  
Three Dimensional ◽  
Mode Shapes ◽  
Thickness Variation ◽  
Rectangular Plates ◽  
Transverse Vibrations ◽  
Modes Of Vibration ◽  
Boundary Characteristic

The free transverse vibrations of thin nonhomogeneous rectangular plates of variable thickness have been studied using boundary characteristic orthogonal polynomials in the Rayleigh-Ritz method. Gram-Schmidt process has been used to generate these orthogonal polynomials in two variables. The thickness variation is bidirectional and is the cartesian product of linear variations along two concurrent edges of the plate. The nonhomogeneity of the plate is assumed to arise due to linear variations in Young's modulus and density of the plate material with the in-plane coordinates. Numerical results have been computed for four different combinations of clamped, simply supported and free edges. Effect of the nonhomogeneity and thickness variation with varying values of aspect ratio on the natural frequencies of vibration is illustrated for the first three modes of vibration. Three dimensional mode shapes for all the four boundary conditions have been presented. A comparison of results with those available in the literature has been made.

Vibrations of Rectangular Plates With Nonuniform Elastic Edge Supports

1980 ◽  
Vol 47 (4) ◽  
pp. 891-895 ◽  
Author(s):  
A. W. Leissa ◽  
P. A. A. Laura ◽  
R. H. Gutierrez
Keyword(s):  
Differential Equation ◽  
Free Vibration ◽  
Exact Solutions ◽  
Ritz Method ◽  
Equation Of Motion ◽  
The Other ◽  
Rectangular Plates ◽  
Governing Differential Equation ◽  
Vibration Problems ◽  
Elastic Edge

Two methods are introduced for the solution of free vibration problems of rectangular plates having nonuniform, elastic edge constraints, a class of problems having no previous solutions in the literature. One method uses exact solutions to the governing differential equation of motion, and the other is an extension of the Ritz method. Numerical results are presented for problems having parabolically varying rotational constraints.

scholarly journals Application of the Electronic Differential Analyzer to the Oscillation of Beams, Including Shear and Rotary Inertia

1955 ◽  
Vol 22 (1) ◽  
pp. 13-19
Author(s):  
C. E. Howe ◽  
R. M. Howe
Keyword(s):  
Transverse Shear ◽  
Normal Modes ◽  
Bending Moment ◽  
Accurate Method ◽  
Rotary Inertia ◽  
Lateral Vibration ◽  
Wide Range ◽  
Modes Of Vibration ◽  
Set Up ◽  
Differential Analyzer

Abstract The equations for normal modes of lateral vibration of beams are set up on the electronic differential analyzer. Beam deflections due to transverse shear and rotary-inertia forces are included. The differential analyzer is shown to be a fast and accurate method for solving the problem. Analyzer outputs include mode shape, slope, bending moment, and shear force along the beam. Curves showing the normal-mode frequencies for the first three modes of vibration of a uniform free-free beam are presented for a wide range of transverse shear and rotary-inertia parameters. The electronic differential analyzer also is utilized to solve the problem of lateral vibration of nonuniform beams.

scholarly journals Transverse Vibrations of Clamped and Simply-Supported Circular Plates with Two Dimensional Thickness Variation

2012 ◽  
Vol 19 (3) ◽  
pp. 273-285 ◽  
Author(s):  
N. Bhardwaj ◽  
A.P. Gupta ◽  
K.K. Choong ◽  
C.M. Wang ◽  
Hiroshi Ohmori
Keyword(s):  
Ritz Method ◽  
Normal Modes ◽  
Mode Shapes ◽  
Thickness Variation ◽  
Two Dimensional ◽  
Circular Plates ◽  
Simply Supported ◽  
Modes Of Vibration ◽  
Dimensional Boundary ◽  
Special Case

Two dimensional boundary characteristic orthonormal polynomials are used in the Ritz method for the vibration analysis of clamped and simply-supported circular plates of varying thickness. The thickness variation in the radial direction is linear whereas in the circumferential direction the thickness varies according to coskθ, wherekis an integer. In order to verify the validity, convergence and accuracy of the results, comparison studies are made against existing results for the special case of linearly tapered thickness plates. Variations in frequencies for the first six normal modes of vibration and mode shapes for various taper parameters are presented.

Vibration Frequencies for a Uniform Beam With One End Elastically Supported and Carrying a Mass at the Other End

1975 ◽  
Vol 42 (4) ◽  
pp. 878-880 ◽  
Author(s):  
D. A. Grant
Keyword(s):  
Normal Modes ◽  
Frequency Equation ◽  
The Other ◽  
Concentrated Mass ◽  
Mass Moment ◽  
Wide Range ◽  
Modes Of Vibration ◽  
Mass Moment Of Inertia ◽  
Elastically Supported ◽  
Range Of Values

In this paper the author obtains the frequency equation for the normal modes of vibration of uniform beams with linear translational and rotational springs at one end and having a concentrated mass at the other free end. The eigenfrequencies for the fundamental mode are given for a wide range of values of mass ratio, mass moment of inertia ratios, and stiffness ratios.

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